Cover mode paints white rectangles over the bands; text may still be selectable under the paint. Crop mode shrinks the visible page boxes so typical viewers hide the bands. Neither rewrites the PDF content stream.
Use Cover when you need the original page size. Use Crop when you want smaller page boxes. For uniform margins on all four sides, use PDF Crop instead.
Yes. Header and footer sizes are measured from the visible crop box, so previously cropped PDFs are handled correctly.